I am reading Dummit and Foote (D&F) Section 13.1 Basic Theory of Field Extensions.
I have a question regarding the nature of extension fields.
Theorem 4 (D&F Section 13.1, page 513) states the following (see attachment):
---------------------------------------------------------------------------------------------------------------
Theorem 4. Let be an irreducible polynomial of degree n over a field F and let K be the field . Let . Then the elements
are a basis for K as a vector space over F, so the degree of the extension is n i.e.
. Hence
consists of all polynomials of degree in
------------------------------------------------------------------------------------------------------------------------------
However, when we come to Example 4 on page 515 of D&F we read the following: (see attachment)
(4) Let and which is irreducible by Eisenstein.
Denoting a root of p(x) by we obtain the field
with an extension of degree 3. ... ... etc
------------------------------------------------------------------------------------------------------------------------------
Now my problem is that in Theorem 4 we read
which becomes
in the situation of Example 4
But then in Example 4 we have
???
It seems that in Theorem 4, we have but in Example (4) we have and we do not have equality but only an isomorphism, that is .
In Field theory we seem to prove that an irreducible polynomial has a root in a field that is isomorphic to the actual field that contains the root.
Does what I am saying make sense? Can someone clarify this issue for me?
Peter